George Soros, who built one of the world’s largest fortunes through a famous series of trades, has turned over nearly $18 billion to Open Society Foundations, according to foundation officials, a move that transforms both the philanthropy he founded and the investment firm supplying its wealth.

Now holding the bulk of Mr. Soros’s fortune, Open Society has vaulted to the top ranks of philanthropic organizations, appearing to become the second largest in the U.S. by assets after the Bill and Melinda Gates Foundation, based...